Properties of Fourier series I

The Fourier series is a powerful tool in signal processing and communications, allowing periodic signals to be expressed as sums of sine and cosine functions. A foundational property of the Fourier series is linearity. If we consider two periodic signals, their linear combination results in a new signal whose Fourier coefficients are simply the corresponding linear combinations of the original signals' coefficients. This property is crucial in applications like frequency modulation (FM) radio, where multiple signals can be transmitted over the same channel without interference.

Time-shifting a periodic signal leaves the magnitude of its Fourier coefficients unchanged. This invariance signifies that the essential characteristics of the signal remain intact despite the shift. For example, in radio broadcasting, this property ensures that shifting a signal in time does not alter its quality. Mathematically, if x(t) is shifted by t₀, the new signal x(t−t₀) has Fourier coefficients e^−jωt X(jω), where X(jω)are the original coefficients. The magnitude ∣X(jω)∣ remains unchanged.

Time reversal is another key property where the sequence of a signal's Fourier series coefficients also undergoes time reversal. For a signal x(t), its time-reversed version x(−t) will have Fourier coefficients that are the complex conjugate of the original coefficients, X(−jω). This property is extensively used in digital signal processing, especially in convolution operations, simplifying the mathematical manipulation of signals.

Symmetry in signals also influences their Fourier coefficients. An even signal, which satisfies x(t) = x(−t), has Fourier coefficients that are real and even. Conversely, an odd signal, where x(t) = −x(−t), has purely imaginary and odd coefficients. These symmetry properties help in simplifying the analysis and synthesis of signals.

In summary, the properties of the Fourier series — linearity, time-shifting invariance, time reversal, and symmetry — are fundamental in various applications, particularly in enhancing signal quality and facilitating signal processing tasks in communications and broadcasting.